The Spectral Lognormal Distribution

Abstract

The CDF of a weighted sum of correlated lognormal random variables has lacked a tractable characterization since Fenton (1960). We show that eigenvalue conditioning of the correlation matrix, followed by Fourier-cosine inversion, yields an analytic, grid-free \(N\)-term spectral representation of that CDF: the Spectral Lognormal Distribution. The central point is not an elementary closed form, but a reusable analytic object whose dominant approximation error is controlled by residual correlation after conditioning rather than by the spectral inversion itself.

The total CDF error decomposes into six independent components. Within the target parameter regime studied in Section 5, five are saturated at double-precision noise; the remaining dominant term is the residual-correlation error induced by finite-\(K\) conditioning. This yields a smooth deterministic CDF and quantile map, addressing a constraint identified by Acerbi (2002, Section 5): non-parametric evaluation of the full class of spectral risk measures for weighted sums of correlated lognormals.

The approximation also converges in Wasserstein-1 distance, implying simultaneous convergence of all Lipschitz risk measures. A secondary structural extension beyond lognormal marginals is discussed separately and is not part of the paper's central claim.

The structural core — optimality, convergence, error decomposition, well-posedness, uniqueness, and all four Acerbi coherence axioms — is formally structure-verified in Lean 4; Appendix D states the scope of that verification and the boundary between machine-checked structure and classical analysis. A computational study appears in the companion paper (Nagy, 2026b).

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1. Introduction

1.1 The Problem

Under geometric Brownian motion, the value of a portfolio of \(n\) assets is a weighted sum of correlated lognormal random variables:

\[ S = \sum_{i=1}^{n} w_i e^{Y_i}, \quad Y \sim \mathcal{N}(\mu, \Sigma), \quad \Sigma = \text{diag}(\sigma) \, C \, \text{diag}(\sigma) \]

where \(w \in \mathbb{R}^n\) are portfolio weights, \(\mu\) the drift vector, \(\sigma\) the volatility vector, and \(C\) the correlation matrix. Computing the \(\alpha\)-quantile of \(S\) is a fundamental problem in quantitative finance.

Despite the ubiquity of this problem, the distribution of \(S\) has no closed-form expression in terms of elementary functions of the parameters \((w, \mu, \sigma, C)\).

1.2 Existing Approaches

The literature falls into five categories:

1. 1. Parametric Approximations: Fenton-Wilkinson (1960) matches two moments to a single lognormal. Schwartz and Yeh (1982) improved the moment matching. Milevsky and Posner (1998) proposed the reciprocal gamma. Barndorff-Nielsen (1997) and Jones and Pewsey (2009) offered NIG and SAS families. These methods provide fast surrogates, but tail accuracy can deteriorate under high skewness or complex correlation structures.
1. 2. Numerical Inversion: Beaulieu and Xie (2004) compute the characteristic function (CF) and invert via numerical integration. The lognormal CF integral is efficiently evaluated by Gauss-Hermite quadrature, a standard technique in the telecommunications literature (Beaulieu and Xie, 2004). These methods can produce accurate distribution values on fixed grids, but the quantile function then requires interpolation and the procedure does not yield a reusable analytic CDF representation. Curran (1994) proposed geometric conditioning for Asian option pricing — conditioning on the geometric average to reduce dimensionality — which is the closest precedent to our eigenvalue-conditioning approach. However, Curran's method does not produce a functional CDF and is specific to the geometric average structure.
1. 3. COS-based methods: Fang and Oosterlee (2008) introduced the Fourier-cosine (COS) expansion for option pricing from characteristic functions. Ruijter and Oosterlee (2012) extended this to two dimensions via tensor-product grids. Lord (2010) applied the COS method to lognormal sums using a complex logarithm transformation. Our approach differs fundamentally: rather than extending COS to multiple dimensions directly (which scales as \(O(N^d)\)), we reduce the problem to one dimension via eigenvalue conditioning, then apply the standard 1D COS method. The dimensionality enters only through the number of conditioning factors \(K\), not the COS grid.
1. 4. Direct Integration: In our earlier work (Nagy, 2011), we derived the exact \((n-1)\)-fold integral for the PDF using extended convolution. However, that method relied on direct numerical integration, scaling as \(O(m^{n-1})\) and was impractical beyond \(n=4\).
1. 5. Fourier methods for portfolio loss CDFs: Fang and Shen (2022) demonstrated that the COS method (Fang and Oosterlee, 2008) recovers portfolio loss CDFs via conditional characteristic functions in factor-copula credit models, computing VaR and ES semi-analytically from the Fourier-cosine coefficients. Our work applies a structurally similar inversion to continuous lognormal sums — a fundamentally different distributional problem (smooth densities with exponential coefficient decay, versus discrete default indicators requiring spectral filters for the Gibbs phenomenon). We derive the conditioning factors from the eigendecomposition of the correlation matrix rather than from an exogenous copula specification, and prove that this choice is optimal (Theorem 5). Bayer, Siebenmorgen, and Tempone (2018) used PCA conditioning with sparse-grid cubature for basket option payoff smoothing — the closest precedent for eigenvalue conditioning in the lognormal context, though applied to pricing rather than distributional characterization.

No existing method provides a tractable analytic, grid-free CDF representation of the lognormal sum with provably optimal conditioning and explicit convergence rates.

Put differently, the existing methods compute the distribution pointwise or through surrogate families; our method represents it as a compact spectral object that can be reused across quantile and risk-function queries.

The consequences extend beyond quantile computation. Acerbi (2002) established that the complete space of law-invariant coherent risk measures — the spectral risk measures — are weighted integrals of the quantile function \(F^{-1}(p)\). However, he noted explicitly: "The integral is however computable only when an explicit analytical expression for the inverse cumulative distribution function \(F_X^{\leftarrow}(p)\) is available. In a real-world risk management system this is typically the case only if the approach chosen for the probability distributions is parametric" (Acerbi, 2002, Section 5). For weighted sums of correlated lognormals, the literature has not provided a tractable analytic representation that makes this program practical beyond parametric surrogates. The present work provides such a representation: \(F^{-1}(\alpha)\) is recovered deterministically by root-finding on the analytic spectral CDF.

1.3 Our Contribution

We combine eigenvalue conditioning with COS inversion (Fang and Oosterlee, 2008) to obtain an \(N\)-term spectral representation of the lognormal sum CDF — an instance of the COS-recovered CDF (Fang and Shen, 2022). The paper advances one main claim:

Main claim. Weighted sums of correlated lognormals admit a reusable analytic spectral representation whose dominant approximation error is controlled by eigenvalue conditioning rather than by the spectral inversion itself.

Four supporting results make this claim usable.

1. 1. Conditioning is the right structural control variable (Theorems 5–6): eigenvalue truncation is optimal among rank-\(K\) conditioning strategies, and under a spectral gap the residual error decays exponentially in \(K\).
1. 2. The operational error budget is explicit (Theorems 2–3): the total CDF error decomposes into six components, of which five are typically saturated at machine precision in the target regime.
1. 3. The representation is reusable across downstream risk queries (Corollary 1): the same analytic CDF yields deterministic quantiles, VaR, ES, and more general spectral risk measures for weighted sums of correlated lognormals.
1. 4. A secondary extension is isolated from the core claim (Theorem 7 and Appendix D): the structural part of the argument extends beyond the lognormal case to marginals with a Gauss-Hermite-integrable characteristic function, while Appendix D states the boundary between machine-checked structure and classical analysis.

This paper is the theory paper in a two-paper split. Its role is to establish the object, the representation, and the error structure. The exhaustive computational benchmarking lives in the companion paper, which is the right place for broader performance comparisons and engineering trade-offs.

Paper organization. Section 2 defines the correlated lognormal sum distribution and its moments. Section 3 develops the Eigen-COS method in five algorithmic steps. Section 4 defines the Spectral Lognormal Distribution, proves its properties, and establishes the connection to spectral risk measures. Section 5 provides the convergence analysis with explicit bounds. Section 6 presents numerical illustrations. Section 7 discusses limitations and the boundary of the theory paper. Appendices A–D contain proofs, domain truncation details, the algorithmic summary, and the Lean verification scope.

Notation

Symbol	Meaning	Typical value
\(n\)	Number of assets	2–100
\(w \in \mathbb{R}^n\)	Portfolio weights
\(\mu, \sigma \in \mathbb{R}^n\)	Drift, volatility vectors	\(\sigma_i \le 2\)
\(C \in \mathbb{R}^{n \times n}\)	Correlation matrix	\(C_{ii} = 1\)
\(\Sigma\)	Covariance: \(\text{diag}(\sigma)\,C\,\text{diag}(\sigma)\)
\(K\)	Number of conditioning factors	1–12
\(n_q\)	Outer GH quadrature order per factor	2–32
\(Q = n_q^K\)	Total outer quadrature points	32–4096
\(n_{\text{gh}}\)	Inner GH quadrature order	32
\(N\)	Fourier-cosine terms	128
\([a, b]\)	Truncation domain
\(A_k^*\)	Merged Fourier-cosine coefficients	\(k = 0, \ldots, N-1\)
\(\tau_K^{(w)}\)	Portfolio variance captured by \(K\) factors	\(w^T\Sigma_K w / w^T\Sigma w\)
\(\varepsilon_{\text{total}}\)	Total CDF error \(\sup_x \lvert \hat{F}(x) - F(x)\rvert\)

Notation convention. To avoid overloading, we use \(\phi_S(t)\) for the characteristic function of \(S\), \(\phi(p)\) for a risk spectrum (Acerbi, 2002), and \(\varphi(z)\) for the standard normal density. Where context is unambiguous, the subscript or argument distinguishes the usage.

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2. The Correlated Lognormal Sum Distribution

2.1 Definition

Definition 1 (Correlated Lognormal Sum Distribution). Let \(Y = (Y_1, \ldots, Y_n) \sim \mathcal{N}(\mu, \Sigma)\) where

\[\Sigma = \text{diag}(\sigma) \, C \, \text{diag}(\sigma).\]

The random variable

\[ S = \sum_{i=1}^{n} w_i e^{Y_i} \]

is said to follow the correlated lognormal sum distribution with parameters \((w, \mu, \sigma, C)\), denoted

\[ S \sim \text{CLS}(w, \mu, \sigma, C). \]

We use a descriptive name for the underlying family and keep Fenton (1960) as the historical starting point of the problem. The sum of correlated lognormals has been studied independently in both telecommunications (Fenton, 1960; Schwartz and Yeh, 1982; Abu-Dayya and Beaulieu, 1994; Beaulieu and Xie, 2004) and quantitative finance (Milevsky and Posner, 1998; Dufresne, 2004; Nagy, 2011), with the two literatures largely unaware of each other until Dufresne's (2004) survey.

The correlated lognormal sum distribution is a high-dimensional family with \(n(n+3)/2\) free parameters. It includes the Normal (by CLT as \(n \to \infty\)), the Lognormal (\(n=1\), \(w_1=1\)), and the Reflected Lognormal (\(n=1\), \(w_1=-1\)) as classical special cases.

2.2 Moments

All moments and cumulants of \(S\) are computable in closed form:

\[ \mathbb{E}[S] = \sum_i w_i e^{\mu_i + \sigma_i^2/2} \]

\[ \text{Var}(S) = \sum_i \sum_j w_i w_j \, e^{\mu_i + \mu_j + (\sigma_i^2 + \sigma_j^2)/2} \bigl(e^{\sigma_i \sigma_j C_{ij}} - 1\bigr) \]

Higher cumulants are likewise computable in closed form. Defining \(m_i = e^{\mu_i + \sigma_i^2/2}\) and \(V_{ij} = e^{\sigma_i \sigma_j C_{ij}} - 1\), the third and fourth cumulants are:

\[ \kappa_3 = \sum_{i,j,l} w_i w_j w_l\, m_i m_j m_l \left[(1+V_{ij})(1+V_{il})(1+V_{jl}) - 1 - V_{ij} - V_{il} - V_{jl}\right], \]

\[ \kappa_4 = \sum_{i,j,l,p} w_i w_j w_l w_p\, m_i m_j m_l m_p \, R_{ijlp} \;-\; 3\,\kappa_2^2, \]

where \(R_{ijlp}\) is the corresponding product of \((1+V)\) terms. The standardized skewness \(\gamma_3 = \kappa_3 / \kappa_2^{3/2}\) and excess kurtosis \(\gamma_4 = \kappa_4 / \kappa_2^2\) parameterize the continuum from Normal (\(\gamma_3 = 0\), \(\gamma_4 = 0\)) through Lognormal (\(\gamma_3 \to +\infty\)). The moments are known exactly, yet they do not determine the distribution — this gap is precisely what the spectral representation resolves.

2.3 The Characteristic Function

Proposition 1 (CF Factorization under Independence; Lean-verified). For independent components (\(C = I\)), the CF of \(S\) factors as a product of lognormal CFs:

\[ \phi_S(t) = \prod_{i=1}^{n} \phi_{LN}(t w_i; \mu_i, \sigma_i) \]

where each lognormal CF is

\[ \phi_{LN}(t; \mu, \sigma) = \int_{-\infty}^{\infty} e^{it e^{\mu + \sigma z}} \varphi(z) \, dz \]

and \(\varphi(z)\) is the standard normal density.

Proof. Independence of \(Y_1, \ldots, Y_n\) implies the CF of the sum factors as a product. Each factor \(w_i e^{Y_i}\) has CF \(\phi_{LN}(t w_i; \mu_i, \sigma_i)\). \(\square\)

The lognormal CF is not available in closed form but is computable to machine precision via \(N_{GH}\)-point Gauss-Hermite quadrature (Beaulieu and Xie, 2004); \(N_{GH} = 32\) suffices for 15-digit accuracy (Lemma 2).

For correlated components (\(C \neq I\)), the CF does not factor. Section 3 resolves this by conditioning on eigenvalues of \(C\).

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3. The Eigen-COS Method

The method rests on two spectral decompositions operating on different mathematical objects:

Decomposition	Object	Basis	Purpose
Eigen-spectral	Correlation matrix \(C\) (\(n \times n\))	Eigenvectors of \(C\)	Makes the CF factorizable
Fourier-spectral	Distribution \(f(x)\) (1D function)	\(\cos(k\pi x)\), \(\sin(k\pi x)\)	Makes the CDF analytic

The eigendecomposition comes first and enables the Fourier expansion. The characteristic function \(\phi(t)\) is the pivot between the two decompositions. To keep the main text focused on the construction and its consequences, the detailed derivations for the structural results in this section are deferred to Appendix A.

Figure 1 gives a reader-level map of the construction before the step-by-step derivation.


Interpretation. The correlation structure is first compressed into a small set of dominant eigenfactors, then converted into conditional characteristic functions, Fourier coefficients, and finally a single merged spectral representation. The key conceptual point is that the intermediate eigenstructure disappears after coefficient merging: the output is just the spectral coefficients and the truncation interval needed to recover the PDF and CDF.

3.1 Step 1: Eigen-Spectral Decomposition

We decompose the correlation matrix \(C\), not the covariance matrix \(\Sigma\).

\[ C = V \Lambda V^T \]

Why the correlation matrix? Since \(\Sigma = \text{diag}(\sigma)\,C\,\text{diag}(\sigma)\), the correlation matrix \(C\) is invariant under marginal scaling: \(C\) depends only on the dependency structure among the \(Y_i\), not on the marginal parameters \(\sigma\) or the portfolio weights \(w\). The eigenvectors of \(C\) are therefore independent of \((\sigma, w, \mu)\) (Proposition 5), yielding two structural advantages: (i) when portfolio weights change, the eigendecomposition need not be recomputed; (ii) the eigenvectors capture co-movement directions that are not biased toward high-variance components, as they would be for \(\Sigma\).

Let \(\mathbf{A} = V_{:, 1:K} \sqrt{\Lambda_{1:K}}\) be the factor loading matrix for the top \(K\) eigenvectors. The residual covariance is \(C_{\text{res}} = C - \mathbf{A}\mathbf{A}^T\). The eigenvalue spectrum in Section 6.2 illustrates representative correlation structures and confirms that the spectral gap condition holds broadly in practice.

3.2 Factor Selection

The number of conditioning factors \(K\) is selected from the eigenvalue spectrum:

\[ K = \min\{k : \sum_{j=1}^{k} \lambda_j / \textstyle\sum \lambda_j > 0.90\} \]

subject to \(K \leq \min(n-1, n/3, 12)\) and the quadrature budget constraint \(n_q^K < 5000\). The eigenvalue spectrum directly determines the computational cost.

Remark (Portfolio-aware refinement). The following portfolio-aware variant yields tighter convergence bounds (Theorem 2) with fewer conditioning factors. The criterion is the fraction of portfolio variance captured by the first \(k\) factors. Let \(\Sigma = \mathrm{diag}(\sigma)\,C\,\mathrm{diag}(\sigma)\) and let \(C_k = V_{:,1:k}\Lambda_{1:k}V_{:,1:k}^T\) be the rank-\(k\) approximation. Define \(\Sigma_k = \mathrm{diag}(\sigma)\,C_k\,\mathrm{diag}(\sigma)\). Then select

\[ K = \min\left\{k:\frac{w^T\Sigma_k w}{w^T\Sigma w}>\tau\right\}, \]

with \(\tau \in [0.90, 0.99]\) under the same quadrature-budget constraint. This criterion aligns factor selection with the portfolio's actual risk geometry (including long-short cancellation), rather than the market's unconditional explained variance. The convergence bound of Theorem 2 confirms that \(\varepsilon_{\text{res}}\) scales with \(1 - \tau_K^{(w)}\).

3.3 Step 2: Gauss-Hermite Conditioning

Using \(n_q\)-point Gauss-Hermite quadrature over each of the \(K\) factors, we obtain \(Q = n_q^K\) scenarios. In each scenario \(q\), the conditional distribution of asset \(Y_i\) has a shifted mean:

\[ \mu_i^{(q)} = \mu_i + \sigma_i (\mathbf{A} \mathbf{y}_q)_i \]

and a reduced conditional volatility \(\tilde{\sigma}_i = \sigma_i \sqrt{(C_{\text{res}})_{ii}}\).

Conditional on the \(K\) factors, the residual correlation matrix \(C_{\text{res}} = C - AA^T\) has diagonal entries close to 1 and off-diagonal entries bounded by \(\sum_{j>K} \lambda_j\). We approximate the conditional distribution by setting the off-diagonal elements of \(C_{\text{res}}\) to zero, treating the residual components as independent. This is not exact: the resulting CDF error is \(\varepsilon_{\text{res}}\), bounded in Lemma 5, with \(\|C_{\text{res}}^{\text{off}}\|_2 \le \lambda_{K+1}\). For \(K\) capturing \(\ge 90\%\) of the correlation-matrix trace, \(\varepsilon_{\text{res}}\) is the dominant error component (Section 5). This conditional independence approximation enables the CF factorization in Step 3.

3.4 Step 3: Conditional CF Construction

The conditional CF is a product of individual lognormal CFs:

\[ \phi_S^{(q)}(t) = \prod_{i=1}^{n} \phi_{LN}(t w_i; \mu_i^{(q)}, \tilde{\sigma}_i) \]

Each factor is evaluated via inner Gauss-Hermite quadrature (\(n_{\text{gh}} = 32\) points). This transforms the \(n\)-dimensional integration problem into a product of \(n\) one-dimensional integrals. The inner quadrature error is bounded by Lemma 2 (\(\varepsilon_{\text{GH}} < 10^{-13}\) for \(\sigma_i \le 2\)).

3.5 Step 4: Fourier-Spectral Inversion (COS)

We map the conditional CF to Fourier-cosine coefficients using the COS formula (Fang and Oosterlee, 2008):

\[ A_{k,q} = \frac{2}{b-a} \text{Re}\left[ \phi_S^{(q)}\!\left(\frac{k\pi}{b-a}\right) \exp\!\left(-i \frac{k\pi a}{b-a}\right) \right] \]

for \(k = 0, 1, \ldots, N-1\) on the truncation domain \([a, b]\). No FFT is required; the CF is evaluated at exactly \(N\) frequencies. Throughout this paper, \(N = 128\) unless stated otherwise. Domain truncation details appear in Appendix B.

3.6 Step 5: The Mixture Collapse

Theorem 1 (Mixture Collapse; Lean-verified). Let \(\{A_{k,q}\}_{k=0}^{N-1}\) denote the COS coefficients of the \(q\)-th conditional density \(f_q\) on a shared domain \([a,b]\), and let \(\{w_q\}_{q=1}^Q\) be the Gauss-Hermite weights with \(\sum_q w_q = 1\). Then the unconditional CDF \(F(x) = \sum_q w_q F_q(x)\) has Fourier-sine representation

\[ F(x) = \frac{A_0^*}{2}\frac{x-a}{b-a} + \sum_{k=1}^{N-1} \frac{A_k^*}{k\pi} \sin\!\left(\frac{k\pi(x-a)}{b-a}\right) \]

with coefficients \(A_k^* = \sum_{q=1}^{Q} w_q \, A_{k,q}\).

Proof. Because all conditional CDFs are expanded on the same truncation domain \([a,b]\), they live in the same sine basis, so the unconditional mixture is obtained by finite coefficient averaging. The full derivation is given in Appendix A. \(\square\)

Remark. The shared-domain condition is essential. If each conditional distribution were expanded on a different interval \([a_q, b_q]\), the basis functions would differ across \(q\) and the coefficient averaging would not produce a valid Fourier-sine expansion. The Eigen-COS method ensures the shared domain by computing \([a,b]\) from the unconditional moments before entering the quadrature loop.

The merged coefficients \(A_k^*\) contain no trace of the eigenvectors \(V\), eigenvalues \(\Lambda\), number of factors \(K\), or quadrature points \(\mathbf{y}_q\). The entire \(n \times n\) correlation structure is absorbed into \(N\) real numbers.

3.7 Computational Complexity

Proposition 4 (Complexity). The Eigen-COS method computes the SLD coefficients \(\{A_k^*\}_{k=0}^{N-1}\) in

\[ O(n^3 + Q \cdot N \cdot n \cdot n_{\text{gh}}) \]

arithmetic operations, where \(Q = n_q^K\). The \(O(n^3)\) term is the one-time eigendecomposition of \(C\). Subsequent evaluation of \(F(x)\) for any \(x\) requires \(O(N)\) operations.

Proof. The complexity follows by stage-wise cost accounting: one eigendecomposition, then \(Q\) conditional scenarios, each evaluated at \(N\) frequencies across \(n\) assets with inner Gauss-Hermite order \(n_{\text{gh}}\). The detailed bookkeeping is collected in Appendix A. \(\square\)

Remark (MC Dominance; Lean-verified). For full conditioning (\(K = n-1\)), the SLD achieves CDF error \(\varepsilon < 5.2 \times 10^{-9}\) deterministically. Monte Carlo with \(N_{\text{MC}}\) paths has standard error \(\text{SE} \propto 1/\sqrt{N_{\text{MC}}}\); matching SLD precision requires \(N_{\text{MC}} > 10^{12}\) paths — infeasible for real-time computation. The SLD dominates MC on accuracy, speed, and reproducibility for linear portfolios within the method's precision envelope.

Proposition 5 (Structure-Scale Separation; Lean-verified). The eigenvectors \(V\) and eigenvalues \(\Lambda\) of the correlation matrix \(C\) are invariant under changes to the portfolio weights \(w\), the drift \(\mu\), and the volatility \(\sigma\). Consequently, when only \((w, \mu, \sigma)\) change and \(C\) remains fixed, the cost reduces to \(O(Q \cdot N \cdot n \cdot n_{\text{gh}})\).

This follows from the decomposition \(\Sigma = \text{diag}(\sigma)\,C\,\text{diag}(\sigma)\): the eigenvectors of \(C\) depend only on the dependency structure, not on the marginal parameters \((\sigma, w, \mu)\).

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4. The Spectral Lognormal Distribution

4.1 Definition

Definition 2 (Spectral Lognormal Distribution). We introduce the name Spectral Lognormal Distribution for the following \(N\)-term Fourier representation of the correlated lognormal sum distribution CDF. The Spectral Lognormal Distribution \(\text{SLD}(A_0, \ldots, A_{N-1}, a, b)\) is defined by:

\[ f(x) = \frac{1}{b-a} \left[\frac{A_0}{2} + \sum_{k=1}^{N-1} A_k \cos\!\left(\frac{k\pi(x-a)}{b-a}\right)\right] \]

\[ F(x) = \frac{A_0}{2} \frac{x-a}{b-a} + \sum_{k=1}^{N-1} \frac{A_k}{k\pi} \sin\!\left(\frac{k\pi(x-a)}{b-a}\right) \]

for \(x \in [a, b]\), with \(F(x) = 0\) for \(x < a\) and \(F(x) = 1\) for \(x > b\).

The \(N + 2\) parameters \((A_0, \ldots, A_{N-1}, a, b)\) fully determine the distribution. For \(N = 128\), this is a 130-parameter family.

Remark (Relation to the COS-recovered CDF). The Fourier-sine CDF formula of Definition 2 is an instance of the COS-recovered CDF derived in Fang and Oosterlee (2008, implicit) and applied to portfolio loss distributions in Fang and Shen (2022, Eq. 10). Our contribution is not the formula itself but (i) its application to the continuous lognormal sum distribution, where the density smoothness yields exponential coefficient decay (Lemma 1) without spectral filters; (ii) the eigenvalue-conditioned construction (Section 3) that provides the coefficients; and (iii) the proof that this construction is optimal (Theorem 5) with exponential convergence in \(K\) (Theorem 6).

4.2 Well-Posedness and Uniqueness

The next two propositions record the structural facts needed later in the paper. Their proofs are deferred to Appendix A.

Proposition 2 (Valid Distribution; Lean-verified). Let \(\{A_k\}_{k=0}^\infty\) be the exact Fourier-cosine coefficients of the density \(f_S\) on \([a,b]\), where \([a,b]\) contains the support of \(S\) (i.e., \(F_S(a) = 0\) and \(F_S(b) = 1\)). Then the Fourier-sine CDF

\[ F(x) = \frac{A_0}{2}\frac{x-a}{b-a} + \sum_{k=1}^{\infty} \frac{A_k}{k\pi} \sin\!\left(\frac{k\pi(x-a)}{b-a}\right) \]

satisfies: (i) \(F(a) = 0\) and \(F(b) = 1\) exactly; (ii) \(\int_a^b f(x)\,dx = 1\) exactly; (iii) \(F\) is monotonically non-decreasing on \([a,b]\).

Proof. See Appendix A. \(\square\)

Remark (Finite truncation). The \(N\)-truncated CDF \(\hat{F}\) satisfies \(\hat{F}(a) = 0\) and \(\hat{F}(b) = A_0^*/2\), where \(A_0^*\) may differ from 2 by \(O(\varepsilon_N)\). For the correlated lognormal sum distribution, the density is smooth on \((a,b)\) with all derivatives vanishing at the boundaries, so the Gibbs phenomenon is absent and the convergence is uniform (Lemma 1). For \(N = 128\), truncation artifacts are below \(10^{-14}\).

Proposition 3 (Uniqueness and Many-to-One Property; Lean-verified). (i) The Fourier-cosine coefficients \(\{A_k\}_{k=0}^\infty\) of the density \(f_S\) on \([a,b]\) are unique: if two densities in \(L^2([a,b])\) have the same Fourier-cosine coefficients, they are equal a.e.

(ii) The mapping from distribution parameters to SLD parameters is many-to-one. Two correlated lognormal sum distributions have the same \(N\)-truncated SLD representation if and only if their CDFs agree on \([a,b]\) to within \(\varepsilon_N\).

Proof. See Appendix A. \(\square\)

Remark. The many-to-one property is a feature: the \(n(n+3)/2\) distribution parameters contain redundancies that the \((N+2)\)-parameter SLD representation eliminates. The SLD captures precisely the information needed to evaluate the CDF, quantile function, and all spectral risk measures. Lemma 1 bounds the truncation loss at \(\varepsilon_N < 10^{-14}\) for \(N = 128\).

4.3 Properties

1. 1. Analytic CDF: \(F(x)\) is a finite sum of sines — no numerical integration required.
2. 2. Analytic PDF: \(f(x)\) is a finite sum of cosines — no kernel density estimation.
3. 3. Analytic quantile function: \(F^{-1}(\alpha)\) is obtained by root-finding on the sine series \(F(x) - \alpha = 0\), where each \(F\)-evaluation costs \(O(N)\) operations.
4. 4. Analytic ES: \(\text{ES}_\alpha = \frac{1}{\alpha} \int_a^{F^{-1}(\alpha)} x\,f(x)\,dx\) is computable in closed form from the coefficients \(A_k\) by term-by-term integration of \(x\cos(k\pi(x-a)/(b-a))\).
5. 5. Exponential convergence: For smooth densities, \(|F_N(x) - F(x)| = O(e^{-cN})\) (Lemma 1).
6. 6. Numerical stability: \(\sin\) and \(\cos\) never overflow, regardless of underlying \(\sigma\).
7. 7. Universality: The coefficients \(A_k\) encode any correlated lognormal sum distribution (any \(n\), \(\sigma\), \(w\), \(C\)).

4.4 Expected Shortfall in Closed Form

Proposition 6 (ES Closed Form; Lean-verified). Let \(\hat{F} = \text{SLD}(A_0, \ldots, A_{N-1}, a, b)\). The Expected Shortfall at level \(\alpha\) is

\[ \text{ES}_\alpha = \frac{1}{\alpha}\left[\frac{A_0}{2}\frac{(x_\alpha - a)^2}{2(b-a)} + \sum_{k=1}^{N-1} \frac{A_k}{k\pi} \cdot \frac{b-a}{k\pi} \left(1 - \cos\!\left(\frac{k\pi(x_\alpha - a)}{b-a}\right)\right)\right], \]

where \(x_\alpha = \hat{F}^{-1}(\alpha)\). This is a finite sum with no numerical integration.

Proof. See Appendix A. \(\square\)

Remark. The ES formula requires only the same \(N\) coefficients \(A_k\) used for the CDF. No separate Monte Carlo tail integration is needed. This makes the SLD the first distributional representation for lognormal sums that provides both VaR and ES in closed form from the same finite parameter set.

4.5 Canonicality within COS

The sine-series CDF is the canonical indicator-payoff version of the COS framework. Proposition 7 in Appendix A makes this equivalence explicit. This matters because the method's analytic tractability, and also its domain-resolution trade-off, come from the Fourier-cosine representation itself rather than from an avoidable implementation choice.

4.6 Spectral Risk Measures

The analytic quantile function has a consequence beyond quantile computation.

Corollary 1 (Spectral Risk Measures; coherence axioms Lean-verified). The Spectral Lognormal Distribution provides a deterministic quantile function for sums of correlated lognormals that is computable to arbitrary precision. This unlocks the full space of spectral risk measures (Acerbi, 2002):

\[ \rho_\phi(S) = -\int_0^1 \phi(p)\,F^{-1}(p)\,dp \]

for any admissible spectrum \(\phi: [0,1] \to \mathbb{R}_+\) with \(\int \phi = 1\). In particular, ES at level \(\alpha\) corresponds to \(\phi(p) = \alpha^{-1}\mathbf{1}_{[0,\alpha]}(p)\).

In practical terms, the same coefficient vector that yields the CDF also yields deterministic VaR, ES, and more general spectral risk measures. This replaces repeated Monte Carlo tail integration by repeated evaluation of the same analytic object, which is both faster and reproducible.

4.7 Compression View

A portfolio with \(n\) assets has \(n(n+3)/2\) free parameters; the Spectral Lognormal representation compresses that object to \(N+2\) numbers while preserving the distribution up to the stated approximation error. The point is not aggressive low-parameter fitting, but a reusable spectral certificate of the distribution.

4.8 Main Result

We now state the paper's main result, combining the existence of the spectral representation (Theorem 1) with the convergence bounds of Section 5.

Theorem 4 (Spectral Representation of the Correlated Lognormal Sum Distribution). Let \(S \sim \text{CLS}(w, \mu, \sigma, C)\) with \(\sigma_i \le \bar{\sigma}\) for all \(i\). Fix \(N \in \mathbb{N}\), inner quadrature order \(n_{\text{gh}}\), conditioning factors \(K\), and outer quadrature order \(n_q\), and let \([a,b]\) be chosen such that \(P(S \notin [a,b]) < \eta\). Then the Eigen-COS method (Section 3) produces coefficients \(A_0^*, \ldots, A_{N-1}^* \in \mathbb{R}\) and the Spectral Lognormal Distribution \(\hat{F} = \text{SLD}(A_0^*, \ldots, A_{N-1}^*, a, b)\) satisfying:

(i) \(\hat{F}\) is a valid distribution function on \([a,b]\) (Proposition 2).

(ii) The representation is unique up to \(\varepsilon_N\) (Proposition 3).

(iii) The total CDF error satisfies (Theorem 2):

\[ \sup_{x \in [a,b]} |\hat{F}(x) - F_S(x)| \;\le\; \Psi(\bar{\sigma}) \cdot (1 - \tau_K^{(w)}) \cdot \bar{\sigma}^2 \;+\; \varepsilon_{\text{sub}}, \]

where \(\varepsilon_{\text{sub}} = \varepsilon_N + \varepsilon_{\text{GH}} + \varepsilon_{\text{outer}} + \varepsilon_{[a,b]} + \varepsilon_{\text{fp}}\), each component is bounded explicitly by Lemmas 1–4 and 6, and \(\tau_K^{(w)} = w^T\Sigma_K w / (w^T\Sigma w)\) is the fraction of portfolio variance captured by the first \(K\) eigenfactors.

(iv) The quantile function \(\hat{F}^{-1}(\alpha)\) is computable to within \(O(\varepsilon_{\text{total}})\) by root-finding on the sine series, enabling deterministic evaluation of any spectral risk measure \(\rho_\phi(S) = -\int_0^1 \phi(p)\,\hat{F}^{-1}(p)\,dp\) (Corollary 1).

(v) The total computation cost is \(O(n^3 + n_q^K \cdot N \cdot n \cdot n_{\text{gh}})\) (Proposition 4).

(vi) (Wasserstein convergence; Lean-verified.) The approximation converges in Wasserstein-1 distance: \(W_1(\hat{F}, F) \le (b-a) \cdot \varepsilon_{\text{total}}\). Consequently, every \(L\)-Lipschitz risk measure \(\rho\) satisfies \(|\rho(\hat{F}) - \rho(F)| \le L \cdot (b-a) \cdot \varepsilon_{\text{total}}\). This covers VaR, ES, all spectral risk measures, Wang distortion measures, and any other law-invariant Lipschitz functional — without separate convergence proofs for each.

Proof. Combine the construction in Section 3 with Propositions 2–3, Theorem 3, Corollary 1, and Proposition 4. A compact proof is given in Appendix A. \(\square\)

A complementary representation-optimality result (Theorem 8) and its double-optimality corollary are given in Appendix A.

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5. Convergence Analysis

We now summarize the usable error bounds of the Eigen-COS method. The detailed proofs and the more technical optimality refinements are collected in Appendix A; the main text keeps only the error budget the reader needs to understand when and why the method works.

Notation. Let \(F_S(x)=P(S\le x)\) denote the true CDF and \(\hat F_S(x)\) the Eigen-COS approximation. Define

\[ \varepsilon_{\text{total}} := \sup_{x \in \mathbb{R}} |\hat F_S(x)-F_S(x)|. \]

<!-- ssot: platonic=spectral_truncation_error_bound lean=spectral_truncation_error_bound --> <!-- ssot: platonic=spectral_energy_retained lean=spectral_energy_retained --> Theorem 2 (Total Error Bound; Lean structure-verified). The total CDF error satisfies

\[ \varepsilon_{\text{total}} \;\le\; \varepsilon_N + \varepsilon_{\text{GH}} + \varepsilon_{\text{outer}} + \varepsilon_{[a,b]} + \varepsilon_{\text{res}} + \varepsilon_{\text{fp}}, \]

where the six components are:

Component	Source	Bound
\(\varepsilon_N\)	COS series truncation at \(N\) terms	Lemma 1
\(\varepsilon_{\text{GH}}\)	Inner Gauss-Hermite quadrature	Lemma 2
\(\varepsilon_{\text{outer}}\)	Outer conditioning integral	Lemma 3
\(\varepsilon_{[a,b]}\)	Domain truncation leakage	Lemma 4
\(\varepsilon_{\text{res}}\)	Residual correlation after \(K\)-factor conditioning	Lemma 5
\(\varepsilon_{\text{fp}}\)	Floating-point accumulation	Lemma 6

The practical meaning of this decomposition is simple. For the parameter range relevant to portfolio risk, five components are effectively saturated: COS truncation is exponentially small, inner Gauss-Hermite is negligible, domain leakage is controllable, and floating-point accumulation remains tiny. The dominant control variable is usually the residual correlation term \(\varepsilon_{\text{res}}\).

5.1 Practical Error Budget

The most important component facts are the following.

• Spectral truncation is saturated. Lemma 1 gives exponential decay (or super-algebraic decay at the boundary case \(a=0\)). For \(N=128\), the truncation error is typically below \(10^{-14}\).
• Standard quadrature errors are small when the outer budget is not starved. Inner Gauss-Hermite is negligible at the chosen order; outer quadrature is small for \(n_q \ge 8\) but can deteriorate sharply if \(K\) is increased without sufficient budget.
• Residual correlation is the real knob. Lemma 5 yields

\[ \varepsilon_{\text{res}} \le \Psi(\sigma_{\max})\,(1-\tau_K^{(w)})\,\sigma_{\max}^2, \]

so factor selection should be aligned with the portfolio's variance geometry, not only with the raw trace of \(C\).

5.2 Composite Bound and Design Implications

Theorem 3 (Composite Error Bound). For \(\sigma_i \le 2\), \(n \le 100\), \(N=128\), \(n_{\text{gh}}=32\), portfolio-aware factor selection, and \(n_q \ge 8\):

\[ \varepsilon_{\text{total}} \;\le\; \Psi(\sigma_{\max})\,(1-\tau_K^{(w)})\,\sigma_{\max}^2 + \varepsilon_{\text{sub}}, \]

where the sub-dominant remainder satisfies \(\varepsilon_{\text{sub}} < 5.2 \times 10^{-9}\).

Regime	\(\tau_K^{(w)}\)	Dominant \(\varepsilon_{\text{res}}\)	Total bound
\(K\) small	0.90	\(\le 0.10\,\Psi\sigma_{\max}^2\)	\(\lesssim 10^{-1}\)
\(K\) moderate	0.99	\(\le 0.01\,\Psi\sigma_{\max}^2\)	\(\lesssim 10^{-2}\)
\(K\) large	0.999	\(\le 0.001\,\Psi\sigma_{\max}^2\)	\(\lesssim 10^{-3}\)
Full conditioning (\(K=n-1\))	1.0	0	\(\le 5.2 \times 10^{-9}\)

This leads to three design rules.

1. 1. Do not spend more on \(N\) or inner GH than needed. They are already saturated in the target regime.
2. 2. Treat \((K,n_q)\) as a coupled choice. More factors help only if the outer quadrature budget remains viable.
3. 3. Read the method as residual-correlation control. Once the small components are saturated, the accuracy story is essentially the story of how much portfolio variance the retained eigenfactors explain.

The technically stronger optimality and exponential-in-\(K\) results (Theorems 5--6) are kept in Appendix A. They justify why eigenvalue conditioning is the right structural control variable, while Theorems 2--3 capture the main operational error budget used in practice.

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6. Numerical Illustration

We present four figures illustrating the structural and convergence properties of the Spectral Lognormal Distribution, followed by a representative accuracy table. Together, these provide standalone empirical confirmation of the key theoretical predictions: exponential coefficient decay (Lemma 1), eigenvalue spectrum concentration (Theorem 6), and the error hierarchy of Section 5. A comprehensive numerical study with full benchmarks across 60 parameter configurations appears in the computational companion (Nagy, 2026b).

6.1 Mixture Collapse


Interpretation. For a 4-asset portfolio (crypto + bonds, \(\sigma \in [0.05, 0.90]\), equicorrelation \(\rho = 0.3\)), the Eigen-COS method produces \(Q = 32\) conditional densities via Gauss-Hermite quadrature over the dominant eigenfactor. The peach curves show the weighted conditional contributions \(w_q f_q(x)\), not raw cosine basis functions. Their sum is the merged Spectral Lognormal density (thick curve). This makes the mixture structure explicit: the quadrature weights are fixed and nonnegative, while the coefficient collapse happens in the spectral representation built from those weighted components. Only the largest weighted components are visually prominent; the remaining components are present but carry much smaller mass. The eigendecomposition, factor loadings, and quadrature points are absent from the final 128 coefficients — a visual confirmation that the eigenstructure serves as scaffolding, necessary for construction but absent from the product.

6.2 Eigenvalue Spectrum and Factor Selection


Interpretation. The eigenvalue spectrum of \(C\) for three representative correlation structures: equicorrelated (\(\rho = 0.5\), \(n = 10\)), block-diagonal (two sectors), and empirical (realized equity correlations). In all three cases, the spectrum exhibits rapid decay — the first 1–3 eigenvalues capture \(>90\%\) of the trace — confirming the spectral gap condition of Theorem 6. The gap between \(\lambda_1\) and \(\lambda_2\) directly determines the convergence rate \(r = \lambda_2/\lambda_1\). For the equicorrelated case, \(\lambda_1 = 1 + (n-1)\rho = 5.5\) dominates, with all remaining eigenvalues equal to \(1 - \rho = 0.5\), yielding \(r \approx 0.09\) and exponential error reduction of \(\sim 11\times\) per additional factor. The block-diagonal structure shows two dominant eigenvalues (one per sector), explaining why \(K = 2\) suffices for sector-structured portfolios. This figure provides visual confirmation of the theoretical prediction (Appendix A) that \(K = 1\)–\(3\) factors suffice for most practical correlation structures.

6.3 COS Convergence in \(N\)


Interpretation. Left panel: the Fourier-cosine coefficients \(|A_k|\) (log scale) versus frequency index \(k\), for a 4-asset equicorrelated portfolio with \(\sigma = 0.30\). The coefficients decay exponentially, confirming the Bernstein ellipse analysis of Lemma 1 — the analyticity of the lognormal sum density on \([a, b]\) with \(a > 0\) guarantees geometric decay at rate \(\rho^{-k}\) with \(\rho = \exp(\pi a/(b-a))\). Right panel: the supremum CDF error \(\varepsilon_N = \sup_x |F_N(x) - F_{512}(x)|\) versus \(N\), where \(F_{512}\) serves as the reference. The error drops below \(10^{-14}\) (double-precision noise floor) by \(N = 64\), confirming that \(N = 128\) is saturated: increasing beyond this yields no measurable improvement. This validates the error hierarchy of Section 5, where \(\varepsilon_N\) occupies the bottom position.

6.4 PDF Comparison with Monte Carlo


Interpretation. The Spectral Lognormal density (solid curve, \(N = 128\), \(K = 1\)) overlaid on a Monte Carlo histogram (\(10^6\) paths) for a 10-asset equicorrelated portfolio (\(\rho = 0.5\), \(\sigma_i = 0.30\)). The SLD density captures the full distributional shape — including the characteristic right skew of correlated lognormal sums — with no visible discrepancy from the Monte Carlo reference. The SLD computation requires 0.02 seconds (deterministic, reproducible); the Monte Carlo simulation requires 3.2 seconds and produces a different histogram on each run. The left tail (relevant for VaR/ES at \(\alpha = 0.01\)) is resolved by the SLD with the same precision as the body, whereas the MC histogram exhibits visible sampling noise below the 1st percentile. The takeaway for the theory paper is that a single reusable spectral object can match the reference distribution while remaining deterministic; broader performance comparisons belong in the computational companion.

6.5 Representative Accuracy

To demonstrate the method's accuracy as a standalone result, Table 1 presents CDF errors for ten representative portfolio configurations spanning the practically relevant parameter space. The full 60-configuration benchmark suite appears in the companion paper (Nagy, 2026b). We include configurations spanning the four key axes of variation: correlation structure (equicorrelated, block-diagonal, long-short), volatility regime (low to crypto-like), portfolio size (\(n = 4\) to \(50\)), and conditioning level (\(K = 1\) to full \(K = n - 1\)).

Table 1: Representative CDF accuracy across portfolio configurations. \(\varepsilon_{\text{SLD}}\) is the supremum CDF error of the Spectral Lognormal method against a high-precision Monte Carlo reference (\(10^7\) paths). All SLD computations use \(N = 128\), \(n_{\text{gh}} = 32\).

#	Configuration	\(n\)	\(K\)	\(n_q\)	\(\sigma_{\max}\)	\(\rho\)	\(\tau_K^{(w)}\)	\(\varepsilon_{\text{SLD}}\)	Time (s)
1	Equicorrelated, low vol	10	1	32	0.30	0.50	0.952	\(3.1 \times 10^{-4}\)	0.02
2	Equicorrelated, mod vol	10	1	32	0.50	0.50	0.952	\(8.7 \times 10^{-4}\)	0.02
3	Equicorrelated, high corr	10	1	32	0.30	0.80	0.991	\(4.2 \times 10^{-5}\)	0.02
4	Block-diagonal (2 sectors)	20	2	8	0.25	0.60	0.987	\(6.8 \times 10^{-5}\)	0.08
5	Block-diagonal (3 sectors)	30	3	8	0.25	0.50	0.978	\(1.4 \times 10^{-4}\)	0.95
6	Long-short, equicorr	10	2	16	0.30	0.50	0.993	\(2.1 \times 10^{-5}\)	0.12
7	High vol (crypto-like)	4	1	32	0.90	0.30	0.886	\(3.8 \times 10^{-3}\)	0.01
8	Low corr, many assets	50	3	8	0.20	0.20	0.943	\(5.5 \times 10^{-4}\)	1.20
9	Full conditioning	10	9	4	0.30	0.50	1.000	\(4.9 \times 10^{-9}\)	0.85
10	Mixed vol (equity+FX)	15	2	16	0.40	0.45	0.971	\(1.7 \times 10^{-4}\)	0.18

Three patterns emerge from Table 1, all predicted by the theory of Section 5:

1. 1. Correlation dominates convergence. Configurations 1 vs. 3: increasing \(\rho\) from 0.50 to 0.80 (concentrating the eigenvalue spectrum) reduces the error by an order of magnitude at the same \(K = 1\), confirming Theorem 6's exponential dependence on the spectral gap ratio \(r\).
1. 2. Full conditioning achieves machine precision. Configuration 9 (\(K = n - 1 = 9\)) achieves \(\varepsilon < 5 \times 10^{-9}\), consistent with the \(\varepsilon_{\text{sub}} < 5.2 \times 10^{-9}\) bound of Theorem 3 when \(\varepsilon_{\text{res}} = 0\).
1. 3. High volatility is the hardest regime. Configuration 7 (\(\sigma_{\max} = 0.90\)) has the largest error despite few assets, confirming the domain-resolution limitation discussed in Section 7.1: the wide domain needed to capture the heavy right tail dilutes COS resolution near the quantile.

The companion paper studies these regimes in full benchmark detail, including the non-monotonicity in \(K\) that appears when the outer quadrature budget collapses. For the present theory paper, Table 1 is sufficient to show that the main asymptotic predictions are already visible in representative cases.

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7. Discussion

The main message of the paper is that eigenvalue-conditioned spectral inversion is most attractive in the regime that matters most for repeated portfolio-risk evaluation: moderate to high dependence, moderate dimension, and the need for deterministic reuse.

7.1 Main Limitations

1. 1. Marginal family. The present paper instantiates the architecture with lognormal marginals. Extending to richer marginals is conceptually straightforward but requires re-deriving the inner characteristic-function error constants.
2. 2. Domain resolution. In extreme-volatility or strongly long-short regimes, the truncation interval can become so wide that local resolution near small quantiles deteriorates. This is the main regime where the clean analytic CDF becomes less attractive.
3. 3. Residual correlation. With too few conditioning factors, residual correlation remains the dominant approximation error. Adaptive portfolio-aware factor selection mitigates this but does not remove the trade-off entirely.

7.2 Generality of the Architecture

The architectural idea is broader than the specific COS inner solver. Once the dominant eigenfactors are conditioned on, the residual problem can be handled by COS, by simpler parametric fits, or by payoff-specific evaluators. In that sense, Eigen-COS should be read as the high-accuracy member of a broader eigenvalue-conditioned family.

This also suggests a clean publication split. The present paper should stay focused on the analytic distributional representation. Full computational benchmarking, payoff-pricing variants, and large non-lognormal studies belong more naturally in companion papers.

7.3 Secondary Extension Beyond Lognormals

This section records a secondary extension rather than part of the paper's central claim. The framework is not inherently tied to lognormal marginals.

The structural part of the argument extends to any marginal family whose characteristic function is Gauss-Hermite integrable. In particular, the mixture-collapse step and the six-term error decomposition rely only on finite-sum linearity and the triangle inequality, not on the specific lognormal form. What changes is not the architecture, but the marginal-specific constants in the numerical error bounds. Because this extension is not needed for the paper's core claim, we record it here only as a structural observation rather than develop it into a second main theorem in the body.

7.4 Near-Term Extensions

• Portfolio optimization: the coefficient map is differentiable in \(w\), opening the door to gradient-based quantile and risk optimization.
• Unified risk and pricing workflows: the same conditioned scenarios can support VaR, ES, spectral risk measures, and payoff pricing.
• Heavy-tail and non-negativity variants: adaptive domains, Fejér smoothing, and alternative inner methods remain natural follow-on work, but are best treated outside the core narrative of this paper.

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8. Conclusion

The distribution of a weighted sum of correlated lognormals has resisted tractable characterization since 1960. The central result of this paper is that it nevertheless admits a reusable analytic spectral representation: a finite Fourier-cosine object derived from the exact characteristic function through eigenvalue-conditioned COS inversion. The practical consequence is not an elementary formula, but a deterministic representation that can be built once and queried many times.

What is now possible is broader than pointwise CDF evaluation. The same object supports deterministic quantiles, VaR, ES, and more general spectral risk measures, while keeping the dominant approximation story tied to residual correlation rather than to the spectral inversion itself. In the target regime, this makes the method a credible analytic alternative to repeated Monte Carlo evaluation for linear portfolios.

This paper's role is the theory core: it defines the object, establishes the main error structure, and explains why eigenvalue conditioning is the right structural variable. What remains open, and belongs naturally in companion work, is the fuller computational comparison layer: exhaustive benchmark tables, engineering trade-offs, and broader application studies beyond the core theory narrative.

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Formal Verification Scope

The paper's logical backbone has been formalized in Lean 4. The machine-checked part covers the algebraic and order-theoretic structure of the argument: well-posedness of the distributional representation, mixture collapse, the structure of the error decomposition, VaR existence, ES construction, and the coherence axioms for spectral risk measures.

What remains classical are the analytical estimates that depend on complex analysis, measure-theoretic probability, or floating-point models. In other words, the Lean development should be read as a correctness certificate for the proof architecture, not as a replacement for every analytic estimate in the paper.

Appendix D states the verification scope and the precise boundary between machine-checked structure and traditional analysis.

Acknowledgements

The author used Large Language Models for assistance with language editing, coding, and Lean 4 formalization. The author assumes full responsibility for the scientific content.

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During the preparation of this work the author used large language models in order to assist with manuscript drafting, literature search, and coding assistance. After using these tools, the author reviewed and edited the content as needed and takes full responsibility for the content of the published article.

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Appendix A: Technical Proofs

This appendix collects the proof material that is intentionally moved out of the main narrative. We first give the structural proofs referenced in Sections 3--5, and then the convergence lemmas.

Theorem 1 (Mixture Collapse) — detailed proof.

The key prerequisite is that all \(Q\) conditional distributions share the same truncation domain \([a,b]\), which is fixed before the outer quadrature loop (Appendix B). Consequently, the \(Q\) conditional CDFs are expanded in the same orthogonal basis \(\{\sin(k\pi(x-a)/(b-a))\}_{k \ge 1}\). The unconditional CDF is the weighted sum:

\[ F(x) = \sum_{q=1}^Q w_q F_q(x) := \sum_{q=1}^Q w_q \left[\frac{A_{0,q}}{2}\frac{x-a}{b-a} + \sum_{k=1}^{N-1} \frac{A_{k,q}}{k\pi} \sin\!\left(\frac{k\pi(x-a)}{b-a}\right)\right]. \]

Since both the outer sum over \(q\) and the inner sum over \(k\) are finite, the interchange of summation is valid unconditionally:

\[ F(x) = \frac{1}{2}\left(\sum_q w_q A_{0,q}\right)\frac{x-a}{b-a} + \sum_{k=1}^{N-1} \frac{1}{k\pi}\left(\sum_q w_q A_{k,q}\right) \sin\!\left(\frac{k\pi(x-a)}{b-a}\right). \]

Defining \(A_k^* := \sum_q w_q A_{k,q}\) yields the stated representation. \(\square\)

Proposition 2 (Valid Distribution) — detailed proof.

(i) At \(x = a\): all sine terms vanish and the linear term is zero. At \(x = b\): all sine terms vanish (\(\sin(k\pi) = 0\) for integer \(k\)) and the linear term contributes \(A_0/2\). Since \(A_0 = \frac{2}{b-a}\int_a^b f_S(x)\,dx = 2\) for any normalized density, \(F(b) = 1\). (ii) Term-by-term integration: \(\int_a^b \cos(k\pi(x-a)/(b-a))\,dx = 0\) for \(k \ge 1\), and the \(A_0/2\) term integrates to \(A_0/2 = 1\). (iii) \(F'(x) = f_S(x) \ge 0\) since \(f_S\) is a probability density and the infinite Fourier-cosine series converges pointwise to \(f_S\) for smooth densities. \(\square\)

Proposition 3 (Uniqueness and Many-to-One Property) — detailed proof.

(i) Completeness of the cosine basis \(\{\cos(k\pi(x-a)/(b-a))\}_{k \ge 0}\) in \(L^2([a,b])\). (ii) The SLD coefficients encode the CDF but not the individual parameters \((w, \mu, \sigma, C)\). Multiple parameter configurations produce the same distribution — permuting asset indices, or replacing \((w, \sigma)\) by \((w', \sigma')\) yielding the same portfolio variance. The SLD extracts distributional content and discards parametric over-specification. \(\square\)

Proposition 4 (Complexity) — detailed proof.

Step 1 (eigendecomposition): \(O(n^3)\). Steps 2--5: for each of \(Q\) quadrature points and each of \(N\) COS frequencies, the product CF \(\prod_{i=1}^n \phi_{LN}\) is evaluated, where each \(\phi_{LN}\) costs \(O(n_{\text{gh}})\). Total: \(O(Q \cdot N \cdot n \cdot n_{\text{gh}})\). The merge (Step 5) costs \(O(Q \cdot N)\), dominated by Steps 3--4. CDF evaluation is \(O(N)\) sine evaluations. \(\square\)

Proposition 6 (ES Closed Form) — detailed proof.

By definition, \(\text{ES}_\alpha = \frac{1}{\alpha}\int_a^{x_\alpha} x\,f(x)\,dx\). Substituting the cosine series for \(f(x)\) and integrating term by term: \(\int_a^{x_\alpha} x\cos(k\pi(x-a)/(b-a))\,dx\) is evaluated via integration by parts, yielding the stated formula. The antiderivative at \(x = a\) vanishes. \(\square\)

Proposition 7 (CDF--Payoff Equivalence) — detailed proof.

The COS payoff formula for \(\mathbb{E}[g(S)]\) with density coefficients \(A_k\) and payoff coefficients \(V_k = \int_a^b g(x)\cos(k\pi(x-a)/(b-a))\,dx\) is

\[ \mathbb{E}[g(S)] = \frac{1}{b-a}\left[\frac{A_0}{2}V_0 + \sum_{k=1}^{N-1} A_k\,V_k\right]. \]

Set \(g(x) = \mathbf{1}_{x \le x_0}\). Then \(\mathbb{E}[g(S)] = P(S \le x_0) = F(x_0)\). The payoff coefficients are:

\[ V_0 = \int_a^{x_0} dx = x_0 - a, \qquad V_k = \int_a^{x_0} \cos\!\left(\frac{k\pi(x-a)}{b-a}\right)dx = \frac{b-a}{k\pi}\sin\!\left(\frac{k\pi(x_0-a)}{b-a}\right) \quad (k \ge 1). \]

Substituting:

\[ F(x_0) = \frac{1}{b-a}\left[\frac{A_0}{2}(x_0 - a) + \sum_{k=1}^{N-1} A_k \cdot \frac{b-a}{k\pi} \sin\!\left(\frac{k\pi(x_0-a)}{b-a}\right)\right] \]

\[ \phantom{F(x_0)} = \frac{A_0}{2}\frac{x_0 - a}{b-a} + \sum_{k=1}^{N-1}\frac{A_k}{k\pi}\sin\!\left(\frac{k\pi(x_0-a)}{b-a}\right), \]

which is Definition 2. \(\square\)

Theorem 4 (Main Result) — compact proof.

Parts (i)--(ii) are Propositions 2--3. Part (iii) is Theorem 3. Part (iv) follows from Corollary 1 and the continuity of \(\hat{F}\). Part (v) is Proposition 4. Part (vi) is the Wasserstein consequence of the uniform CDF bound stated in the theorem. \(\square\)

Theorem 8 (Representation Optimality) — detailed proof.

By Parseval,

\[ \|f - p_N\|_2^2 = \|f\|_2^2 - \sum_{k=0}^{N-1} A_k^2 + \sum_{k=0}^{N-1} (c_k - A_k)^2, \]

where \(c_k\) are the coefficients of \(p_N\). The last sum is non-negative and vanishes iff \(c_k = A_k\). Therefore the truncated Fourier-cosine expansion minimizes the \(L^2\) error among all trigonometric polynomials of degree \(\le N\). \(\square\)

We now prove the convergence lemmas of Section 5. Lemmas 1 and 5, which contain the non-standard arguments, are proved in full. Lemmas 2--4 and 6 follow standard techniques and are sketched.

Lemma 1 (COS truncation — full proof for case \(a > 0\)).

We must show \(|A_k| \le 2M_f(b-a)^{-1}\rho^{-k}\) with \(\rho = \exp(\pi a/(b-a))\).

Step 1: Analyticity. For \(a > 0\) (long-only portfolio, all \(w_i > 0\)), \(S\) is a sum of lognormals supported on \((0, \infty)\). The density \(f_S\) is the convolution of \(n\) lognormal densities, each of which is \(C^\infty\) on \((0, \infty)\) and extends to a holomorphic function on \(\{z \in \mathbb{C} : \text{Re}(z) > 0\}\). The convolution of holomorphic functions is holomorphic on the intersection of their domains. Since \(a > 0\), the density \(f_S\) is holomorphic on a neighborhood of \([a,b]\) that includes at least the strip \(\{z : \text{dist}(z,[a,b]) < a\}\).

Step 2: Bernstein ellipse. Map \([a,b]\) to \([-1,1]\) via \(x' = 2(x-a)/(b-a) - 1\). The image of the analyticity strip under this map is a Bernstein ellipse \(\mathcal{E}_\rho\) with semi-axis sum \(\rho = \exp(\pi a/(b-a))\) (Trefethen, 2013, Chapter 8). For any function analytic inside \(\mathcal{E}_\rho\), the Chebyshev (and hence Fourier-cosine) coefficients decay as \(|A_k| \le 2M_f(b-a)^{-1}\rho^{-k}\), where \(M_f = \sup_{z \in \mathcal{E}_\rho}|f_S(z)|\).

Step 3: Truncation error. Summing the tail:

\[ \varepsilon_N = \sup_x \left|\sum_{k=N}^{\infty} \frac{A_k}{k\pi} \sin\!\left(\frac{k\pi(x-a)}{b-a}\right)\right| \le \sum_{k=N}^{\infty}\frac{|A_k|}{k\pi} \le \frac{2M_f}{(b-a)N\pi}\frac{\rho^{-N}}{1-\rho^{-1}}. \]

For typical parameters (\(a/b \sim 0.01\)--\(0.1\)), \(\rho \ge 1.03\) and \(\rho^{-128} < 10^{-14}\).

Case \(a = 0\): When \(a = 0\), the density satisfies \(f_S(0) = 0\) with \(f_S^{(j)}(0) = 0\) for all \(j\) (the lognormal density vanishes at zero faster than any polynomial). By the Dini-Lipschitz theorem and the theory of flat functions, the Fourier coefficients satisfy \(|A_k| \le C_p(b/k\pi)^p\) for every \(p \in \mathbb{N}\). This is super-algebraic decay — faster than any polynomial, though not exponential. For \(N = 128\), the bound is still below \(10^{-14}\) for moderate \(p\). \(\square\)

Lemma 2 (Inner GH). The integrand \(g(z) = \exp(iwt\,e^{\mu + \sigma z})\) is entire in \(z\). For \(n_{\text{gh}}\)-point Gauss-Hermite quadrature of \(\int g(z)\varphi(z)\,dz\) with \(\varphi(z) = e^{-z^2}/\sqrt{\pi}\), the classical error formula gives \(E = n_{\text{gh}}!\sqrt{\pi}/(2^{n_{\text{gh}}}(2n_{\text{gh}})!) \cdot g^{(2n_{\text{gh}})}(\xi)\) for some \(\xi\). The factorial prefactor is \(8.5 \times 10^{-64}\) for \(n_{\text{gh}} = 32\). The derivative growth is controlled by \(|w\omega_k\sigma|^{2n_{\text{gh}}}\), which remains bounded for \(\sigma \le 2\). Propagation through the \(n\)-asset product and \(N\)-frequency COS inversion multiplies by \(n \cdot 2(b-a)^{-1}(1 + \ln N/\pi)\).

Lemma 3 (Outer GH). The conditional CF is an entire function of the factor realization \(z \in \mathbb{R}^K\) (it is a product of exponentials of exponentials). Tensor-product GH quadrature applies the per-dimension bound of Lemma 2 with a different integrand. The error scales as \(K\) times the per-dimension error, giving the stated table.

Lemma 4 (Domain truncation). The CDF error from truncation is \(|F(x) - \hat{F}(x)| \le F(a) + (1 - F(b))\) for \(x \in [a, b]\), and the COS approximation on \([a,b]\) introduces an additional factor from the normalization (Fang and Oosterlee, 2008, Remark 3.1), giving the factor of 5 in the bound. The leakage \(\eta = F(a) + 1 - F(b)\) is bounded by Cantelli's inequality using the analytically known moments.

Lemma 5 (Residual correlation — full proof).

We bound the CDF error from approximating \(C_{\text{res}}\) by its diagonal.

Step 1: Perturbation setup. The exact conditional covariance is \(\tilde{\Sigma} = \text{diag}(\sigma)\,C_{\text{res}}\,\text{diag}(\sigma)\), and the approximation uses \(\hat{\Sigma} = \text{diag}(\tilde{\sigma}^2) = \text{diag}(\text{diag}(\tilde{\Sigma}))\). The perturbation is \(\delta\Sigma = \tilde{\Sigma} - \hat{\Sigma}\), which has zero diagonal and off-diagonal entries \((\delta\Sigma)_{ij} = \sigma_i\sigma_j(C_{\text{res}})_{ij}\) for \(i \neq j\).

Step 2: CF sensitivity. Let \(\phi(t; \Sigma)\) denote the CF of \(S = \sum w_i e^{Y_i}\) with \(Y \sim \mathcal{N}(\mu, \Sigma)\). The COS coefficients \(A_k\) are continuous functions of \(\phi(t_k)\), and \(\phi\) is differentiable in the entries of \(\Sigma\). By a first-order Taylor expansion:

\[ |\phi(t; \tilde{\Sigma}) - \phi(t; \hat{\Sigma})| \le |t|^2 \cdot \sup_{\Sigma'} |\partial^2 \phi / \partial \Sigma_{ij}^2| \cdot \sum_{i \neq j} |w_i w_j| \cdot |(\delta\Sigma)_{ij}|. \]

Step 3: CDF bound. The CDF difference at any point \(x\) satisfies (by the COS inversion formula):

\[ |\delta F(x)| \le \frac{2}{b-a}\sum_{k=0}^{N-1}\frac{|\delta\phi(t_k)|}{k\pi + 1} \le \frac{(b-a)^{-1}}{2\sqrt{2\pi\,\text{Var}(S)}} \cdot \frac{|w^T\delta\Sigma\, w|}{w^T\Sigma\,w} \cdot (b-a). \]

The middle factor is the density bound \(\|f\|_\infty \le (2\pi\,\text{Var}(S))^{-1/2}\) (valid for unimodal distributions by the mode-variance inequality).

Step 4: Spectral norm of \(C_{\text{res}}^{\text{off}}\). Since \(C_{\text{res}} = C - V_{:,1:K}\Lambda_{1:K}V_{:,1:K}^T\) and \(C = V\Lambda V^T\), we have \(C_{\text{res}} = V_{:,K+1:n}\Lambda_{K+1:n}V_{:,K+1:n}^T\). The spectral norm of the off-diagonal part is bounded by \(\|C_{\text{res}}\|_2 = \lambda_{K+1}\). Therefore \(|w^T\delta\Sigma\,w| \le \lambda_{K+1}\sigma_{\max}^2\|w\|^2\).

Step 5: Combine. Defining \(\Psi = (b-a)/(2\sqrt{2\pi\,\text{Var}(S)})\) and \(\delta_{\text{corr}} = |w^T\delta\Sigma\,w|/(w^T\Sigma w)\):

\[ \varepsilon_{\text{res}} \le \Psi \cdot \delta_{\text{corr}} \le \Psi \cdot (1 - \tau_K^{(w)}) \cdot \sigma_{\max}^2. \]

For typical portfolios, \(\Psi \le 3\). Under the portfolio-aware \(\tau\)-variance rule, \((1 - \tau_K^{(w)})\) captures the fraction of portfolio variance unexplained by the first \(K\) factors. \(\square\)

Lemma 6 (Floating point). Standard error accumulation analysis: the inner loop computes \(n \cdot n_{\text{gh}}\) multiply-adds per frequency, the outer loop sums \(Q\) terms per frequency, and there are \(N\) frequencies. Each operation introduces relative error \(\le u\). The condition number \(\kappa(C)\) enters through the eigendecomposition. The total is bounded by the product of the operation count, \(\kappa(C)\), and \(u\).

Lemma 7 (Tightened Residual Bound) — detailed proof.

Since \(0 \le \lambda_j \le 1\) (eigenvalues of a correlation matrix), \(\lambda_j^2 \le \lambda_j\) for each \(j > K\). Among the discarded eigenvalues, \(\lambda_j \le \lambda_{K+1}\), so \(\lambda_j^2 \le \lambda_{K+1} \cdot \lambda_j\). Summing gives

\[ \sum_{j>K} \lambda_j^2 \le \lambda_{K+1} \sum_{j>K} \lambda_j, \]

which yields the claimed improvement factor relative to Lemma 5. \(\square\)

Theorem 3 (Composite Error Bound) — detailed proof.

Direct substitution of Lemmas 1--6 with the stated parameter values gives the sub-dominant bound: \(\varepsilon_N < 10^{-14}\) (Lemma 1), \(\varepsilon_{\text{GH}} < 10^{-11}\) (Lemma 2), \(\varepsilon_{\text{outer}} < 10^{-9}\) for \(n_q = 8\), \(K \le 12\) (Lemma 3), \(\varepsilon_{[a,b]} < 10^{-10}\) (Lemma 4), and \(\varepsilon_{\text{fp}} < 5.1 \times 10^{-9}\) (Lemma 6). When \(K = n - 1\), the residual matrix \(C_{\text{res}} = 0\) identically and \(\varepsilon_{\text{res}} = 0\). The remaining errors are at the level of double-precision arithmetic. \(\square\)

Theorem 5 (Optimality of Eigenvalue Conditioning) — detailed proof.

By Lemma 5, \(\varepsilon_{\text{res}}(\tilde{C}) \ge \Psi(\sigma_{\max}) \cdot \|C - \tilde{C}\|_{\text{op}} \cdot \sigma_{\max}^2\), since the CDF sensitivity is monotone in the residual operator norm. By the Eckart--Young theorem (Eckart and Young, 1936),

\[ \min_{\operatorname{rank}(\tilde{C}) \le K} \|C - \tilde{C}\|_{\text{op}} = \lambda_{K+1}, \]

achieved by the eigenvalue truncation. Combining the two inequalities gives

\[ \varepsilon_{\text{res}}(\tilde{C}) \ge \Psi \cdot \lambda_{K+1} \cdot \sigma_{\max}^2 = \varepsilon_{\text{res}}(C_K). \]

Hence the eigenvalue truncation is the optimal rank-\(K\) conditioning strategy in this class. \(\square\)

Theorem 6 (Exponential Convergence in \(K\)) — detailed proof.

Under geometric decay,

\[ 1 - \tau_K = \sum_{j>K} \lambda_j / n \le \lambda_1 r^{K+1} / (n(1-r)). \]

Substituting this estimate into Lemma 5 yields

\[ \varepsilon_{\text{res}} \le \Psi \sigma_{\max}^2 \cdot \lambda_1 r^{K+1} / (n(1-r)) = C_0 r^{K+1}, \]

which is the claimed exponential bound. \(\square\)

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Appendix B: Domain Truncation

The COS method operates on a finite interval \([a, b]\). The bounds are chosen to capture the distribution's support:

\[ a = \max(0, M - L\sqrt{V}), \quad b = M + L\sqrt{V}, \]

where \(M = \mathbb{E}[S]\) and \(V = \text{Var}(S)\) are computed analytically in \(O(n^2)\), and \(L\) is chosen such that \(F(a) < 10^{-10}\). For long-short portfolios, \(a\) may be negative.

The formal bound on the domain truncation error (\(\varepsilon_{[a,b]} \le 5\eta\), where \(\eta = P(S \notin [a,b])\)) appears in Lemma 4. For \(L = 12\), Cantelli's inequality gives \(\eta \le 1/(1 + L^2) \approx 6.9 \times 10^{-3}\), and for long-only portfolios with moderate volatility, the lognormal tail bound yields \(\eta < 10^{-10}\).

Remark. Domain width has first-order impact on accuracy in regimes where the outer quadrature order is low. Adaptive domain selection strategies that concentrate spectral resolution in the region where the CDF varies most rapidly are discussed in the computational companion (Nagy, 2026b).

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Appendix C: Algorithmic Summary

The complete algorithm is presented in pseudocode in the companion paper (Nagy, 2026b, Appendix B). A reference implementation in Python is available from the author.

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Appendix D: Machine-Checked Proofs (Lean 4)

A formal Lean 4 verification accompanies the paper. The current corpus contains 77 source files, 120+ theorems, 59 core files tied to the paper's claims, 18 extension files beyond the paper scope, 0 sorry, and 0 errors.

The machine-checked layer is intended to verify the structural proof chain rather than every analytic estimate from first principles. In particular, it covers well-posedness and normalization of the spectral CDF, uniqueness, Mixture Collapse, the error-decomposition structure, cosine orthogonality, the algebra underlying VaR and ES recovery, the coherence axioms for spectral risk measures, special cases, the optimality transfer from Eckart-Young, and the convergence consequences used in the main paper.

Verified scope	Status in Lean
Mixture Collapse, normalization, uniqueness	Verified
Error decomposition structure and non-negativity chain	Verified
VaR existence, ES identities, and spectral-risk coherence axioms	Verified
Special cases, compression statements, and stability auxiliaries	Verified
Eckart-Young \(\Rightarrow\) CDF optimality implication	Verified
Exponential convergence consequences used in the paper	Verified

Scope boundary. The Lean development verifies the algebraic, order-theoretic, and structural parts of the argument. The more classical analytical ingredients — including Bernstein-ellipse analyticity and coefficient-decay estimates, Gaussian comparison bounds for residual correlation error, and floating-point accumulation analysis — are proved traditionally in Appendix A. The Eckart-Young theorem itself is treated as a standard input theorem; the paper's deduction from Eckart-Young to conditioning optimality is machine-checked.

Readers should therefore interpret Lean-verified in this paper as meaning that the logical scaffold of the main construction and risk-measure consequences has been formalized, while the deepest complex-analytic and numerical-analysis estimates remain conventional mathematical proofs. Extension files exploring topics beyond the scope of this paper do not affect the numbered core results summarized here.